The structure of network materials is stochastic. This chapter introduces the minimum set of geometric parameters required to describe the network structure. This set includes the fiber and crosslink densities, the mean segment length, a measure of preferential fiber orientation, and the connectivity index. The relation between the mean segment length and the fiber density is established for two- and three-dimensional networks with cellular and fibrous architectures. The effect of fiber tortuosity, fiber preferential alignment, and excluded volume interactions on the mean segment length are outlined. The statistics of pore sizes in networks of fibrous and cellular types is discussed in terms of the geometric network parameters. The percolation threshold, at which the first connected path forms across the network domain, is discussed for specific methods used to generate the network.

Many Network materials exhibit time-dependent behavior. This chapter begins with a review of essential results from viscoelasticity. Further, the mechanisms leading to network scale time dependence are analyzed in three separate sections. The influence of the fiber material time dependence on the network behavior is discussed first. In networks embedded in a fluidic matrix, the migration of the fluid in and out of the network may produce time dependent mechanical behavior. The basic notions of poroelasticity are presented and the conditions under which this mechanism becomes important for the network-scale mechanics are outlined. Time dependence is also produced by nonbonded fiber interactions. This is an essential component of the mechanics of thermal molecular networks and a section is devoted to this topic. In networks in which crosslinks are transient, such as ionomers and vitrimers, material behavior is strongly time dependent, and a section is dedicated to this issue. Multiple mechanisms act concurrently in applications and identifying their individual contributions is not always easy. A discussion aimed to assist the interpretation of experimental data is included.

Various types of fibers encountered in Network materials are presented and classified in this chapter. Their mechanical behavior is of primary concern here. The first section describes the structure and mechanical behavior of cellulose fibers, polymeric fibers used in nonwovens, and collagen fibers forming connective tissue. The remainder of the chapter is divided into three parts presenting the mechanical behavior of athermal fibers, thermal filaments, and of fiber bundles. The linear, nonlinear, and rupture characteristics of athermal fibers are presented. Thermal filaments, which form molecular networks such as elastomers and gels, are described by the Gaussian, Langevin, and self-avoiding random walk models. Models describing the mechanics of semiflexible filaments are presented. The section on the mechanics of fiber bundles presents a number of results relevant for bundles of continuous and discontinuous (staple) fibers, including the effect of bundle twisting and of packing on the axial stiffness and strength of the bundle. These results apply to many networks of practical importance which are composed from fiber bundles.

A review of current constitutive formulations for Network materials is presented in this chapter. Network materials are composed from discrete elements and are not continua. Their behavior is somewhat similar to that of mechanisms. Furthermore, deformation is generally nonaffine due to the stochastic network structure. These observations render difficult the adaptation of classical constitutive equations for this class of materials. These issues are discussed in detail in the opening section. Further, the chapter is divided into four sections, each presenting models of a certain type. The first category includes phenomenological models defined based on a free energy functional and examples relevant for thermal networks (elastomers and gels) are presented. The next three categories encompass mechanism-based models, which are divided based on the degree to which the respective models account for nonaffinity in affine, quasi-affine, and nonaffine models. An outline of the challenges and opportunities related to the development of mechanism-based constitutive models for Network materials is presented in closure.

This introductory chapter defines the class of Network materials and provides numerous examples. Network materials are classified based on several criteria. They are divided into thermal and athermal, as a function of the dependence of the fiber behavior on temperature, in dry, embedded or embedding, as a function of the presence of embedded entities (nanoparticles, macromolecular entities) or of an embedding matrix, in crosslinked and non-crosslinked function of the nature of fiber interactions. This classification is used in the book to define categories of mechanical behavior. The chapter closes with an outline of the book.

This chapter is dedicated to composites in which the reinforcement is a stochastic fiber network. Many network materials are reinforced by the addition of fillers of various geometry. However, in most current applications, filler dimensions are orders of magnitude larger than the characteristic length scales of the network. The focus of this chapter is on the properties of composites with matched filler-network length scales. The four sections of the chapter present the mechanics of networks reinforced with particles of dimensions comparable with the network pores, networks reinforced with stiff fibers, interpenetrating networks in which reinforcement is provided by the interaction with another network which spans the same spatial domain, and of networks embedded in a continuum matrix. It is shown that exceptional properties may be achieved due to the emergence of interphases in thermal and athermal networks with rigid fillers, and in interpenetrating network systems. The results and concepts presented are aimed to stimulate the future development of reinforced network materials.

Affine models have been used traditionally to describe the deformation of networks. Due to their prevalence, this chapter is dedicated to the review of such formulations. The chapter begins with a brief review of finite kinematics of continua and the definition of stress measures. Further, the affine deformation is defined and several parameters used to quantify the degree of nonaffinity are introduced. An expression is derived to quantify the evolution of preferential fiber orientation during affine deformation. Several constitutive models based on the affine deformation assumption are discussed: The affine models for molecular networks of flexible and semi-flexible filaments, and the affine model for athermal networks. The stress–optical law is reviewed, and its relation to the affine deformation models is discussed.

This chapter presents results related to the interaction of fibers. In networks, fibers form crosslinks and develop contacts. The mechanical behavior of the crosslinks between athermal fibers and between thermal filaments is discussed, with emphasis on crosslink failure and strength. Contacts between fibers form either along the fiber length, within bundles of parallel fibers, or at specific sites where fibers cross at an angle. The mechanics of elastic contacts with and without cohesive interactions between fibers is reviewed. Sliding of fibers in contact is often encountered in networks and friction has strong effects on network mechanics. Several results related to friction at elastic contacts are reviewed.